Two integrals that involve quadratic equations in the sine and cosine functions and are
defined as:
C(x) = ∫ cos (πt2/2) dt and
C(y) = ∫ sin (πt2/2) dt, integrated from 0 to x.
They are quite frequently used in optics studying → Fresnel diffraction
and similar topics. The Fresnel integrals are also used in railway and freeway constructions.
These integrals may be evaluated to arbitrary precision using
→ power series.
Alternatively the amplitudes may be found graphically by use of
→ Cornu's spiral.

A technique in spectroscopy for recording a spectrum from
each point of an extended object. The field of view image is divided into a
multitude of small components using different methods, e.g. lenslet arrays, fiber
bundles, or image slicers. From each component a spectrum is extracted or an image is
reconstructed at a particular wavelength.

The integral admitted by the equations of a body of infinitesimal
mass moving under the → gravitational attractions
of two massive bodies, which move in circles about their
→ center of gravity. The Jacobi integral is the only
known conserved quantity for the circular
→ restricted three-body problem.
In the co-rotating system it is expressed by the equation:
(1/2) (x·2 +
y·2
+ z·2) = U - CJ,
where the dotted coordinates represent velocities,
U is potential energy, and CJ the constant of
integration (→ zero-velocity surface).
The Jacobi integral has been used for two different purposes:
1) to construct surfaces of zero velocity which limit the regions
of space in which the small body, under given initial conditions,
can move, and 2) to derive a criterion
(→ Tisserand's parameter) for re-identification
of a → comet whose orbit has suffered severe perturbations
by a planet. Also known as Jacobi constant.

Named after Karl Gustav Jacobi (1804-1851), a German mathematician
who did important work on elliptic functions, partial differential
equations, and mechanics; → integral.

multiple integral

درستال ِ بستایی

dorostâl-e bastâyi

Fr.: intégrale multiple

A series of successive integrations in which the integral operator acts on the
result of preceding integration.